I'm working on a statistical model which involves many degrees of freedom $i=1...S$. Each degree of freedom is described by a gamma distribution with its own parameters, which we will assume to be all different $v_i \sim \Gamma(k_i,\theta_i)$ with $k_i >0$, $\theta_i >0$
I want to calculate the joint pdf of all the variables, given their sum:
$$ P(\vec{v}|V) = \frac{1}{Z_V} \big [ \prod_{i=1}^S P_i(v_i) \big ] \delta( \sum_{i=1}^{S}v_i-V )$$
Now i calculate the partition function of the model:
$$ Z_V = \frac{1}{ \prod_{i=1}^S \Gamma(k_i) \theta_i^{k_i}} \int \prod_{i=1}^{S} d v_i v_i^{k_i-1} e^{-\frac{v_i}{\theta_i}} \delta(\sum_{i=1}^{S} v_i-V) $$
Using the Fourier representation of the delta function to split the integrals, one gets :
$$ Z_V \propto \int \prod_{i=1}^{S} d v_i v_i^{k_i-1} e^{-\frac{v_i}{\theta_i}} \int_{-\infty}^{\infty} dt \ e^{ it (\sum_{i=1}^{S} v_i-V)} = $$
$$ \int_{-\infty}^{\infty} dt e^{-itV}\int \prod_{i=1}^{S} d v_i v_i^{k_i-1} e^{-\frac{v_i}{\theta_i}} \ e^{ it v_i } $$
Each term in the productory is the moment generating function of a Gamma, and i end up with this integral
$$ \int_{-\infty}^{\infty} dt \frac{1}{\prod_{i=1}^{S} (1-i \theta_i t)^{k_i}}e^{-itV} $$
I can not go further with the problem. I tried several theoretical physics tricks, like Feynman and Schwinger parameterization, but it did not helped. Alternatively, one may consider it as the Fourier transform of the product of the generating functions and try to apply the residues theorem, but the $k_i$s are not integers.
Anyone has an idea about how to tackle this integral?